The Astrophysical Journal, 706:417–435, 2009 November 20 doi:10.1088/0004-637X/706/1/417
C
2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
A MODEL FOR THE WAVEFORM BEHAVIOR OF ACCRETING MILLISECOND X-RAY PULSARS: NEARLY
ALIGNED MAGNETIC FIELDS AND MOVING EMISSION REGIONS
Frederick K. Lamb
1,2
, Stratos Boutloukos
1,3
, Sandor Van Wassenhove
1
, Robert T. Chamberlain
1
,KaHoLo
1
,
Alexander Clare
1
, Wenfei Yu
1
, and M. Coleman Miller
3
1
Center for Theoretical Astrophysics and Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street,
Urbana, IL 61801-3080, USA; fkl@illinois.edu
2
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801-3074, USA
3
Department of Astronomy and Maryland Astronomy Center for Theory and Computation, University of Maryland, College Park, MD 20742-2421, USA
Received 2008 August 31; accepted 2009 October 5; published 2009 October 30
ABSTRACT
We investigate further a model of the accreting millisecond X-ray pulsars we proposed earlier. In this model, the
X-ray-emitting regions of these pulsars are near their spin axes but move. This is to be expected if the magnetic
poles of these stars are close to their spin axes, so that accreting gas is channeled there. As the accretion rate and the
structure of the inner disk vary, gas is channeled along different field lines to different locations on the stellar sur-
face, causing the X-ray-emitting areas to move. We show that this “nearly aligned moving spot model” can explain
many properties of the accreting millisecond X-ray pulsars, including their generally low oscillation amplitudes
and nearly sinusoidal waveforms; the variability of their pulse amplitudes, shapes, and phases; the correlations in
this variability; and the similarity of the accretion- and nuclear-powered pulse shapes and phases in some. It may
also explain why accretion-powered millisecond pulsars are difficult to detect, why some are intermittent, and why
all detected so far are transients. This model can be tested by comparing with observations the waveform changes
it predicts, including the changes with accretion rate.
Key words: pulsars: general – stars: neutron – stars: rotation – X-rays: bursts – X-rays: stars
1. INTRODUCTION
Highly periodic millisecond X-ray oscillations have been de-
tected with high confidence in 22 accreting neutron stars in
low-mass X-ray binary systems (LMXBs), using the Rossi X-
ray Timing Explorer (RXTE) satellite (see Lamb & Boutloukos
2008). We refer to these stars as accreting millisecond X-ray
pulsars (AMXPs). Accretion-powered millisecond oscillations
have so far been detected in 10 AMXPs. They are always ob-
servable in seven AMXPs, but are only intermittently detected
in three others. Nuclear-powered millisecond oscillations have
been detected with high confidence during thermonuclear X-
ray bursts in 16 AMXPs. Persistent accretion-powered mil-
lisecond oscillations have been detected in two AMXPs that
produce nuclear-powered millisecond oscillations; intermittent
accretion-powered millisecond oscillations have been detected
in two others.
The AMXPs have several important properties:
Low oscillation amplitudes. The fractional amplitudes of
the accretion-powered oscillations of most AMXPs are often
only ∼1%–2%.
4
Persistent accretion-powered oscillations with
amplitudes 1% are often detected with high confidence in IGR
J00291+5934 (Galloway et al. 2005; Patruno 2008) and XTE
J1751−305 (Markwardt et al. 2002; Patruno 2008). Persistent
accretion-powered oscillations with amplitudes as low as 2% are
regularly seen in XTE J1807−294 (Zhang et al. 2006; Chou et al.
2008; Patruno 2008), XTE J0929−314 (Galloway et al. 2002),
and XTE J1814−338 (Chung et al. 2008; Patruno 2008). The
4
We characterize the strengths of oscillations by their rms amplitudes
because the rms amplitude can be defined for any waveform, is usually
relatively stable, and is closely related to the power. We convert reported
semi-amplitudes of purely sinusoidal oscillations or Fourier components to
rms amplitudes by dividing by
√
2.
amplitude of the accretion-powered oscillation seen in SWIFT
1756.9−2508 was ∼6% (Krimm et al. 2007). The intermittent
accretion-powered oscillations detected in SAX J1748.9−2021
(Gavriil et al. 2007; Altamirano et al. 2008; Patruno 2008),
HETE J1900.1−2455 (Galloway et al. 2007), and Aql X-1
(Casella et al. 2008) have amplitudes ∼0.5%–3%.
Nearly sinusoidal waveforms. The waveforms (light curves)
of the accretion-powered oscillations of most AMXPs are
nearly sinusoidal (see Wijnands 2006 and the references in
the preceding paragraph). The amplitude of the first harmonic
(fundamental) component is usually 10 times the amplitude of
the second harmonic (first overtone) component, although the
ratio can be as small as ∼3.5, as is sometimes the case in XTE
J1807−294 (Zhang et al. 2006), or even 1, as is sometimes
the case in SAX J1808.4−3658 (see, e.g., Hartman et al. 2008).
Highly variable oscillation amplitudes. The fractional ampli-
tudes of the accretion-powered oscillations of most AMXPs vary
in time by factors ranging from ∼2to∼10. Observed fractional
amplitudes vary from 0.7% to 1.7% in SAX J1748.9−2021,
from 0.7% to 3.7% in XTE J1751−305, from 3% to 7% in XTE
J0929−314, from 1% to 9% in IGR J00291+5934, from 2% to
11% in XTE J1814−338, from 1% to 14% in XTE J1808−338,
and from 2% to 19% in XTE J1807−294 (see the references
above).
Highly variable pulse phases. The phases of accretion-
powered pulses have been seen to vary rapidly by as much as
∼0.3 cycles in several AMXPs, including SAX J1808.4−3658
(Morgan et al. 2003; Hartman et al. 2008) and XTE J1807−294
(Markwardt 2004). Wild changes in the apparent pulse fre-
quency have been observed with both signs at the same accretion
rate in XTE J1807−
294 (see Markwardt 2004). If interpreted as
caused by changes in the stellar spin rate, these phase variations
would be more than a factor of 10 larger than expected for the
largest accretion torques and smallest inertial moments thought
417
418 LAMB ET AL. Vol. 706
possible for these systems (see Ghosh & Lamb 1979b; Lattimer
& Prakash 2001).
Undetected accretion-powered oscillations. More than 80 ac-
creting neutron stars in LMXBs are known (Chakrabarty 2005;
Liu et al. 2007), but accretion-powered millisecond X-ray oscil-
lations have so far been detected in only 10 of them. Accretion-
powered oscillations have not yet been detected even in 13
AMXPs that produce periodic nuclear-powered millisecond os-
cillations, indicating that they have millisecond spin periods
(Lamb & Boutloukos 2008); eight of these also produce kilo-
hertz quasi-periodic oscillations (QPOs) with frequency sepa-
rations that indicate that they not only have millisecond spin
periods but also have dynamically important magnetic fields
(Boutloukos & Lamb 2008).
Intermittent accretion-powered oscillations. Accretion-
powered millisecond X-ray pulsations have been detected
only occasionally in SAX J1748.9−2021 (Gavriil et al. 2007;
Altamirano et al. 2008; Patruno 2008), HETE J1900.1−2455
(Galloway et al. 2007), and Aql X-1 (Casella et al. 2008).
When oscillations are not detected, the upper limits are typically
0.5%.
Correlated pulse arrival times and amplitudes. The phase
residuals of the accretion-powered pulses of several AMXPs
appear to be anti-correlated with their fractional amplitudes, at
least over some of the amplitude ranges observed. AMXPs that
show this type of behavior include XTE J1807−294 and XTE
J1814−338 (Patruno 2008).
Similar accretion- and nuclear-powered pulses. The shapes
and phases of the nuclear-powered X-ray pulses of the AMXPs
SAX J1808.4−3658 (Chakrabarty et al. 2003) and XTE
J1814−338 (Strohmayer et al. 2003) are very similar to the
shapes and phases of their accretion-powered X-ray pulses.
Concentration in transient systems of AMXPs with accretion-
powered oscillations. The AMXPs in which accretion-powered
oscillations have been detected tend to be found in binary
systems that have outbursts lasting about a month (but see
Galloway et al. 2008) separated by quiescent intervals lasting
years (Chakrabarty 2005; Riggio et al. 2008). The accretion
rates of these neutron stars are very low.
In this paper, we investigate further the “nearly aligned
moving spot” model of AMXP X-ray emission that we proposed
previously (Lamb et al. 2006, 2007, 2008). This model has three
main features:
1. The strongest poles of the magnetic fields of neutron
stars with millisecond spin periods are located near—and
sometimes very near—the stellar spin axis. This behavior is
expected for several magnetic field evolution mechanisms.
2. The star’s magnetic field channels accreting gas close
to its spin axis, creating X-ray-emitting areas there and
depositing nuclear fuel there.
5
3. The X-ray-emitting areas on the stellar surface move, as
changes in the accretion rate and the structure of the inner
disk cause accreting gas to be channeled along different
field lines to different locations on the stellar surface. (The
magnetic field of the neutron star is fixed in the stellar crust
on the timescales relevant to the phenomena considered
here.)
5
Muno et al. (2002) considered a single bright spot near the spin axis as well
as a uniformly bright hemisphere and antipodal spots near the spin equator as
possible reasons for the nearly sinusoidal waveforms of some X-ray burst
oscillations, but did not consider accretion-powered oscillations or other
consequences of emission from near the spin axis.
These features provide the basic ingredients needed to under-
stand the AMXP properties discussed above. This is the subject
of the sections that follow. As a guide to these sections, we
summarize our results here.
1. Emission from near the spin axis naturally produces weak
modulation, regardless of the viewing direction. The reason
is that uniform emission from a spot centered on the spin
axis is axisymmetric about the spin axis and therefore
produces no modulation. Emission from a spot close to
the spin axis has only a small asymmetry and therefore
produces only weak modulation.
2. Emission from near the spin axis also naturally produces a
nearly sinusoidal waveform, because the asymmetry of the
emission is weak and broad.
3. If the emitting area is close to the spin axis, even a small
movement in latitude can change the oscillation amplitude
by a substantial factor.
4. If the emitting area is close to the spin axis, movement in
the longitudinal direction by a small distance can change
the phase of the oscillation by a large amount.
Changes in the latitude and longitude of the emitting area
are expected on timescales at least as short as the ∼0.1 ms
dynamical time at the stellar surface and as long as the
∼10 day timescale of the variations observed in the mass
accretion rate.
5. If the emitting area is very close to the spin axis and
remains there, the oscillation amplitude may be so low
that it is undetectable. The effects of rapid changes in
the position of the emitting area—possibly in combination
with other effects, such as reduction of the modulation
fraction by scattering in circumstellar gas—may also play
a role in reducing the detectability of accretion-powered
oscillations in neutron stars with millisecond spin periods.
These effects may explain the fact that accretion-powered
X-ray oscillations have not yet been detected in many
accreting neutron stars that are thought to have millisecond
spin periods and dynamically important magnetic fields.
6. If the emitting area is very close to the spin axis, a small
change in the accretion flow can suddenly channel gas
farther from the spin axis, causing the emitting area to
move away from the axis. This can make a previously un-
detectable oscillation become detectable. Temporary mo-
tion of the emitting area away from the spin axis may
explain the intermittent accretion-powered oscillations of
some AMXPs (Lamb et al. 2009).
7. If the pulse amplitude and phase variations observed in
AMXPs are caused by motion of the emitting area, they
should be correlated. In particular, the pulse phase should
be much more scattered when the pulse amplitude is very
low. The reason is that changes in the longitudinal position
of the emitting area by a given distance produce much larger
phase changes when the emitting area is very close to the
spin axis, which is also when the oscillation amplitude is
very low.
The observational consequences discussed so far depend
only on features (2) and (3) of the model, i.e., that the
accretion-powered X-ray emission of AMXPs comes from
areas near their spin axes and that these areas move
significantly on timescales of hours to days.
8. The picture of AMXP X-ray emission outlined here sug-
gests that the shapes and phases of the nuclear- and
accretion-powered pulses are similar to one another in some
AMXPs because the nuclear- and accretion-powered X-ray
No. 1, 2009 MODEL FOR WAVEFORM BEHAVIOR OF AMXPs 419
emission comes from approximately the same area on the
stellar surface. The reason for this is that in some cases,
the mechanism that concentrates the magnetic flux of the
accreting neutron star near its spin axis, as it is spun up, will
naturally produce magnetic fields strong enough to confine
accreting nuclear fuel near the magnetic poles at least par-
tially, even though the dipole component of the magnetic
field is weak.
9. The picture of neutron star magnetic field evolution and
AMXP X-ray emission outlined here also suggests a pos-
sible explanation for why the AMXPs in which accretion-
powered oscillations have been detected are in transient
systems. If most neutron stars in LMXBs were spun up by
accretion from a low spin rate to a high spin rate, their mag-
netic poles were forced very close to their spin axes, making
accretion-powered oscillations difficult or impossible to de-
tect. However, those stars that are now in compact transient
systems now experience infrequent episodes of mass accre-
tion, and the accretion rate is very low. By now they have
been spun down from their maximum spin rates, a pro-
cess that could force their magnetic poles away from their
spin axes enough to produce detectable accretion-powered
oscillations.
These last two observational consequences depend on feature
(1) of the model, i.e., on how the magnetic fields of neutron
stars evolve as they are spun up and down by accretion and
electromagnetic torques.
In the remainder of this paper, we discuss in detail the features
of the model and its observational implications. In Section 2,we
outline our approach, discussing our modeling of X-ray emis-
sion from the stellar surface, our computational and the code
verification methods, and the pulse profile representation we use.
We present our results in Sections 3 and 4. These results
are based on our computations of several hundred million
waveforms for different emitting regions, beaming patterns,
stellar models, and viewing directions. In Section 3, we consider
the shape and amplitude of X-ray pulses as a function of the size
and inclination of the emitting areas, the compactness of the star,
and the stellar spin rate. In Section 4, we consider the changes
in the pulse amplitude and phase produced by various motions
of the emitting regions on the stellar surface and explore the
origins of correlated changes in the pulse amplitude and phase
and the effects of rapid movement of the emitting areas. We also
discuss why oscillations have not yet been detected in many
accreting neutron stars in LMXBs and why accretion-powered
oscillations are detected only intermittently in some AMXPs.
In Section 5, we summarize the results of our model calcula-
tions. We also discuss how the magnetic poles of most AMXPs
can be forced close to their spin axes, how such mechanisms
may explain why the AMXPs that produce accretion-powered
millisecond oscillations are transient pulsars, the consistency of
the model with the observed properties of rotation-powered mil-
lisecond pulsars, and possible observational tests of the model
discussed here.
Further results of our investigation of the present model
will be presented elsewhere (S. Boutloukos et al. 2009, in
preparation).
2. X-RAY WAVEFORM MODELING
2.1. Modeling the X-ray Emission
In the radiating spot model of AMXP X-ray emission, the
waveform seen by a distant observer depends on the sizes,
shapes, and positions of the emitting regions on the stellar
surface; the beaming pattern of the radiation; the compactness,
radius, and spin rate of the star; and the direction from which the
star is observed. The properties of the X-ray-emitting regions are
determined by the strength and geometry of the star’s magnetic
field, the locations where plasma from the accretion disk enters
the magnetosphere, the extent to which the accreting plasma
becomes threaded and channeled by the stellar magnetic field,
and the resulting plasma flow pattern onto the stellar surface.
In principle, accreting plasma can reach the stellar surface
in two basic ways: (1) by becoming threaded by the stellar
magnetic field and then guided along stellar field lines to the
vicinity of a stellar magnetic pole (Lamb et al. 1973;Basko
& Sunyaev 1975; Elsner & Lamb 1976; Ghosh et al. 1977;
Ghosh & Lamb 1979a, 1979b) or (2) by penetrating between
lines of the stellar magnetic field via the magnetic version of
the Rayleigh–Taylor instability (Lamb 1975a, 1975b; Elsner &
Lamb 1976, 1977; Arons & Lea 1976
;Lamb1977) and then
spiraling inward to the stellar surface.
If a centered dipole component is the strongest component
of the star’s magnetic field, plasma in the accretion disk that
becomes threaded and then channeled to the vicinity of a
magnetic pole is expected to impact the star in a partial or
complete annulus around the pole, producing a crescent- or
ring-shaped emitting area near the pole. If the axis of the dipole
field is significantly tilted relative to the spin axis and the spin
axis is aligned with the axis of the accretion disk, a crescent-
shaped emitting region is expected (see, e.g., Basko & Sunyaev
1975, 1976; Ghosh et al. 1977; Daumerie et al. 1996; Miller
1996; Miller et al. 1998; Romanova et al. 2003). If instead the
dipole axis is very close to the spin axis, as in the model of
AMXP X-ray emission proposed here, the emitting region may
completely encircle the spin axis (see, e.g., Ghosh & Lamb
1979a, 1979b; Romanova et al. 2003).
The north and south magnetic poles of some AMXPs may
be very close to the same spin pole, producing a very off-
center dipole moment orthogonal to the spin axis (see Chen &
Ruderman 1993; Chen et al. 1998; and Section 5.1). If so, ac-
creting matter will be channeled close to the spin axis, but may
be channeled preferentially toward one magnetic pole, produc-
ing an emitting region with approximately onefold symmetry
about the spin axis, or about equally toward both poles, produc-
ing an emitting region with approximately twofold symmetry
about the spin axis. In the first case, the first harmonic of the
spin frequency is likely be the dominant harmonic in the X-ray
waveform, whereas in the second case, the second harmonic
is likely to dominate. Which case occurs will depend on the
accretion flow through the inner disk. In either case, the X-ray
emission will come from close to the spin axis.
The neutron stars that are AMXPs may well have even more
complicated magnetic fields, with significant quadrupole and
octopole components. Higher multipole components are likely
to play a more important role in the AMXPs than in the classic
strong-field accretion-powered pulsars, because the magnetic
fields of the AMXPs are much weaker. As a result, accreting
plasma can penetrate closer to the stellar surface, where the
higher multipole moments of the star’s magnetic field have a
greater influence on the channeling of accreting plasma (Elsner
&Lamb1976). In this case, plasma will still tend to be channeled
toward regions on the surface where the magnetic field is
strongest and will tend to impact the surface in rings or annuli,
but the emission pattern may be spatially complex and vary
rapidly in time (Long et al. 2008).
420 LAMB ET AL. Vol. 706
Disk plasma that penetrates between lines of the stellar
magnetic field will continue to drift inward as it loses its angular
momentum, probably predominantly via its interaction with the
star’s magnetic field (Lamb & Miller 2001). Cold plasma will
remain in the disk plane and impact the star in an annulus where
the disk plane intersects the stellar surface (Miller et al. 1998;
Lamb & Miller 2001). If some of the accreting plasma were
to become hot, the forces exerted on it by the stellar magnetic
field would tend to drive it toward the star’s magnetic equator
(Michel 1977), causing it to impact the stellar surface in an
annulus around the star’s magnetic equator. However, emission
and inverse Compton scattering of radiation is likely to keep the
accreting plasma cold (Elsner & Lamb 1984), so that it remains
in the disk plane as it drifts inward. Plasma that penetrates to
the stellar surface via the magnetic Rayleigh–Taylor instability
is likely to impact the stellar surface in rapidly fluctuating,
irregular patterns (see Romanova et al. 2006, 2008).
Whether accreting plasma reaches the stellar surface pre-
dominantly via channeled flow along field lines or via unstable
flow between field lines depends on the accretion rate and the
spin frequency of the star (see Lamb 1989; Romanova et al.
2008; Kulkarni et al. 2008). Under some conditions, plasma
may accrete in both ways simultaneously (see Miller et al. 1998;
Romanova et al. 2008; Kulkarni et al. 2008).
The sizes, shapes, and locations of the emitting areas on the
surface of an accreting magnetic neutron star and the properties
of the emission from these areas are expected to change on
timescales at least as short as the ∼1 ms dynamical timescale
near the star. This expectation is supported by recent simulations
of accretion onto weakly magnetic neutron stars (see Romanova
et al. 2003, 2004, 2006; Long et al. 2008; Romanova et al. 2008;
Kulkarni et al. 2008). However, changes in AMXP X-ray fluxes
can be measured accurately using current instruments only by
combining ∼100–1000 s of data and hence only variations in
waveforms on timescales longer than this can be measured
directly. Consequently, the emitting areas and beaming patterns
that are relevant for comparisons with current observations of
waveforms are the averages of the actual areas and beaming
patterns over these relatively long times. The emitting areas
and beaming patterns that we use in our computations should
therefore be interpreted as averages of the actual areas and
beaming patterns over these times.
We have computed the X-ray waveforms produced by emit-
ting regions with various sizes, shapes, and positions, for several
different X-ray-beaming patterns and a range of stellar masses,
compactnesses, and spin rates. We find that in many cases these
waveforms can be approximated by the waveforms generated
by a circular, uniformly emitting spot located at the centroid of
the emitting region. The main reason for this is that an observer
sees half the star’s surface at a time (or more, when gravitational
light deflection is included), which diminishes the influence of
the size and detailed shape of the emitting region on the wave-
form. Consequently, we focus here on the waveforms produced
by uniformly emitting circular spots. We will discuss the wave-
forms produced by emitting areas with other shapes, such as
rings or crescents, in a subsequent paper (S. Boutloukos et al.
2009, in preparation).
In addition to studying the X-ray waveforms produced by
emitting areas with fixed sizes, shapes, positions, and radiation-
beaming patterns, we are also interested in the changes in
waveforms produced by changes in the these properties of the
emitting areas. The changes we investigate should be understood
as occurring on the timescales 100 s that can be studied using
current instruments. It is not yet possible to compute from first
principles the accretion flows and X-ray emission of AMXPs
on these timescales, so simplified models must be used. (The
simulations referred to earlier follow the accretion flow for a
few dozen spin periods or dynamical times, intervals that are
orders of magnitude shorter than the intervals that are relevant).
In the following sections, we consider radiation from a single
spot, from two antipodal spots, and from two spots in the same
rotational hemisphere, near the star’s spin axis. Although many
uncertainties remain, recent magnetohydrodynamic simulations
of accretion onto weakly magnetic neutron stars have found that
gas impacts 1%–20% of the stellar surface (Romanova et al.
2004), equivalent to the areas of circular spots with angular
radii of 10
◦
–53
◦
. These radii are consistent with analytical
estimates of the sizes of the emission regions of accreting
neutron stars with weak magnetic fields (Miller et al. 1998;
Psaltis & Chakrabarty 1999). Consequently, we focus on spot
sizes in this range.
An observer may see radiation from a single spot either
because the accretion flow pattern strongly favors one pole of
a dipolar stellar magnetic field over the other, or because the
observer’s view of one pole is blocked by the inner disk or
by accreting plasma in the star’s magnetosphere (see McCray
&Lamb1976; Basko & Sunyaev 1976). An observer may
see radiation from two antipodal spots if emission from both
magnetic poles is visible. Finally, an observer may see radiation
from two spots near the same rotation pole if neutron vortex
motion drives both of the star’s dipolar magnetic poles toward
the same rotation pole (see Chen et al. 1998).
To make it easier for the reader to compare cases, we usually
report results for our “reference” star, which is a 1.4 M
star
with a radius of 5 M in units where G = c = 1 (10.3 km for
M = 1.4 M
), spinning at 400 Hz as measured at infinity, but
we also discuss other stellar models. For the same reason, we
usually consider spots with angular radii of 25
◦
. This is not an
important limitation, because the observed waveform depends
only weakly on the size of the emitting spots, as discussed in
Section 3.2. We describe how the results change if the spot is
larger or smaller.
2.2. Computing X-ray Waveforms
The X-ray waveforms calculated here assume that radiation
propagating from emitting areas on the stellar surface reaches
the observer without interacting with any intervening matter.
The bolometric X-ray waveforms that would be seen by a dis-
tant observer were calculated using the Schwarzschild plus
Doppler (S+D) approximation introduced by Miller & Lamb
(1998). The S+D approximation treats exactly the special rela-
tivistic Doppler effects (such as aberrations and energy shifts)
associated with the rotational motion of the stellar surface,
but treats the star as spherical and uses the Schwarzschild
spacetime to compute the general relativistic redshift, trace the
propagation of light from the stellar surface to the observer,
and calculate light travel-time effects. For the stars considered
here, and indeed for any stars that do not both rotate rapidly
and have very low compactness, the effects of stellar oblate-
ness and frame dragging are minimal and are negligible com-
pared to uncertainties in the X-ray emission (see Cadeau et al.
2007).
We describe the emission from the stellar surface using
coordinates centered on the star. When considering emission
from a single spot, we denote the angle between its centroid
No. 1, 2009 MODEL FOR WAVEFORM BEHAVIOR OF AMXPs 421
and the star’s spin axis by i
s
and its azimuth in the stellar
coordinate system by φ
s
. When considering emission from two
spots, we somewhat arbitrarily identify one as the primary spot
and the other as the secondary spot. We denote the inclination
and azimuth of the centroid of the primary spot by i
s1
and φ
s1
and the inclination and azimuth of the centroid of the secondary
spot by i
s2
and φ
s2
. We denote the inclination of the observer
relative to the stellar spin axis by i.
In computing the waveforms seen by distant observers, we
use as our global coordinate system Schwarzschild coordinates
(r, θ , ϕ, t ) centered on the star with θ = 0 aligned with the
star’s spin axis and ϕ = 0 in the plane containing the spin axis
and the observer. We choose the zero of the Schwarzschild time
coordinate t so that a light pulse that propagates radially from
a point on the stellar surface immediately below the observer
(i.e., at θ = i and ϕ = 0) arrives at the observer at t = 0.
We carried out many calculations to test and verify the
computer code used to obtain the results we report here. We
determined that the code was giving sufficiently accurate results
by varying the spatial and angular resolutions used. For most
of the cases considered in this paper, the emitting spots were
sampled by a grid of 250 points in latitude and 250 points in
longitude, the radiation-beaming pattern was specified at 10
4
angles, and the flux seen by a distant observer was computed at
10
4
equally spaced values of the star’s rotational phase. In some
cases, finer grids were used.
We verified the code used here by comparing its results with
analytical and numerical results for several test cases:
1. We tested our code’s representation of emitting areas and
ray tracing in flat space by comparing the results given by
our code with exact analytical results for the absolute flux
seen by an observer directly above uniform, isotropically
emitting circular spots of various sizes. The numerical
results agreed with the analytical results.
2. We tested our code’s computation of special relativistic
Doppler boosts, aberrations, and propagation-time effects
in several ways. We compared the results given by our code
with exact analytical results for the waveforms produced
by emission in (a) a pencil beam normal to the surface
and (b) a thin fan beam tangent to the surface of a rapidly
rotating star. We also compared the results given by our
code with analytical results for the waveforms produced
by a small spot on the surface of a slowly rotating star in
flat space emitting (a) isotropically and (b) in a beaming
pattern representing Comptonized emission (see Poutanen
&Gierli
´
nski 2003). The numerical results agreed with the
analytical results.
3. We tested our code’s computation of the general relativistic
redshift and light deflection for nonrotating stars by (a)
comparing the deflection of a fan beam tangent to the
stellar surface given by our code for a variety of stellar
compactnesses with the analytical expressions for the light
deflection given by Pechenick et al. (1983) and Page
(1995); (b) comparing the absolute flux given by our
code for an observer directly above isotropically emitting
uniform circular spots of various sizes with independent
semi-analytical results for these cases; (c) comparing the
symmetries of the waveform and the phase of the waveform
maximum given by our code with exact analytical results
for these quantities; and (d) comparing the shape of the
waveforms given by our code with the shapes reported
by Pechenick et al. (1983) and Strohmayer (1992). Our
numerical results agreed satisfactorily with the comparison
results in all cases.
6
4. We tested our code’s computation of the waveforms pro-
duced by emission from slowly rotating stars in general
relativity by comparing the rms oscillation amplitudes it
gives with the amplitudes given by the approximate analyt-
ical formulae of Viironen & Poutanen (2004). Where the
results of Viironen & Poutanen are expected to be accurate,
the two sets of rms amplitudes agreed to better than 1%; in
many cases the agreement was much better. We also com-
pared the waveform given by our code for an isotropically
emitting spot inclined 45
◦
from the spin axis of a 1.4 M
star with a Schwarzschild coordinate radius of 5M spinning
at 600 Hz seen by an observer at an inclination of 45
◦
with
the waveform reported by Poutanen & Beloborodov (2006)
for this case; the two waveforms agreed to better than 1%.
Further details of these tests and comparisons will be given in
a subsequent paper (S. Boutloukos et al. 2009, in preparation).
2.3. Constructing Pulse Profiles
The X-ray flux seen by a given observer will evolve continu-
ously in time as the star rotates and the emission from the stellar
surface changes, generating the observed waveform W (t). As
noted in Section 2.1, the accretion flow from the inner disk to
the stellar surface is expected to vary on timescales at least as
short as the ∼1 ms dynamical timescale near the neutron star,
which will cause the sizes, shapes, and positions of the emitting
regions, and therefore the observed waveform, to vary on these
timescales.
The sensitivity of current instruments is too low to measure
the waveform of an AMXP on timescales as short as 1 ms.
However, nearly periodic waveforms with periods this short
can be partially characterized by folding segments of flux data
centered at a sequence of clock times t
i
(see, e.g., Hartman et al.
2008; Patruno 2008). If the data are folded with a period P
f
that is chosen to agree as closely as possible with the local,
approximate repetition period P (t
i
) of the waveform, one can
construct a time sequence of pulse profiles W
P
(φ,t
i
); here φ is
the pulse phase over one cycle.
The pulse profiles W
P
(φ,t
i
) constructed by folding flux data
are averages of the actual pulse profiles over the time interval
required to construct a stable profile, which can be hundreds
or even thousands of seconds, 10
5
–10
6
times longer than the
∼1 ms dynamical timescale near the neutron star. The folded
pulse profiles are therefore likely to vary more slowly and have
less detail than the X-ray waveform, a point to which we will
return in Section 4.
The waveforms of AMXPs can be modeled even on the
dynamical timescale near the stellar surface, but such waveforms
would contain much more information than can be studied using
current observations. Consequently, we focus here on modeling
folded pulse profiles. We define a computed pulse profile as the
waveform seen by a given observer when a star with a constant
6
The waveforms reported by Pechenick et al. (1983) for two antipodal spots
are slightly inaccurate, as shown by the following two tests. The waveform
seen by an observer in the star’s rotation equator viewing two identical
antipodal spots in the rotation equator should be the same at 180
◦
as at 0
◦
,but
this is not the case for their waveform for this geometry (see their Figure 7).
More generally, the flux from a uniform, isotropically emitting, circular spot
on a spherical, nonrotating star should depend only on the angle between the
radius through the center of the spot and the radius to the observer. This is not
quite true for the waveforms reported by Pechenick et al. (1983). The
waveforms given by our code pass these tests (for details, see S. Boutloukos
et al. 2009, in preparation).
